As the swim coach at Springer, Emily selects which athletes will participate in the state-wide swim relay. The relay team swims $\frac{3}{5}$ of a mile in total, with each team member responsible for swimming $\frac{3}{10}$ of a mile. The team must complete the swim in $\frac{3}{4}$ of an hour. How many swimmers does Emily need on the relay team?
Answer: To find out how many swimmers Emily needs on the team, divide the total distance ( $\frac{3}{5}$ of a mile) by the distance each team member will swim ( $\frac{3}{10}$ of a mile). $ \dfrac{{\dfrac{3}{5} \text{ mile}}} {{\dfrac{3}{10} \text{ mile per swimmer}}} = {\text{ number of swimmers}} $ Dividing by a fraction is the same as multiplying by the reciprocal. The reciprocal of ${\dfrac{3}{10} \text{ mile per swimmer}}$ is ${\dfrac{10}{3} \text{ swimmers per mile}}$ $ {\dfrac{3}{5}\text{ mile}} \times {\dfrac{10}{3} \text{ swimmers per mile}} = {\text{ number of swimmers}} $ $ \dfrac{{3} \cdot {10}} {{5} \cdot {3}} = {\text{ number of swimmers}} $ Reduce terms with common factors by dividing the $3$ in the numerator and the $3$ in the denominator by $3$ $ \dfrac{{\cancel{3}^{1}} \cdot {10}} {{5} \cdot {\cancel{3}^{1}}} = {\text{ number of swimmers}} $ Reduce terms with common factors by dividing the $10$ in the numerator and the $5$ in the denominator by $5$ $ \dfrac{{1} \cdot {\cancel{10}^{2}}} {{\cancel{5}^{1}} \cdot {1}} = {\text{ number of swimmers}} $ Simplify: $ \dfrac{{1} \cdot {2}} {{1} \cdot {1}} = {2} $ Emily needs 2 swimmers on her team.